DATA HANDLING 37

4.1 Looking for Information

In your day-to-day life, you might have come across information, such as:

(a) Runs made by a batsman in the last 10 test matches.

(b) Number of wickets taken by a bowler in the last 10 ODIs.

(d) Number of story books read by each of your friends etc.

The information collected in all such cases is called data. Data is usually collected in

the context of a situation that we want to study. For example, a teacher may like to know

the average height of students in her class. To find this, she will write the heights of all the

students in her class, organise the data in a systematic manner and then interpret it

accordingly.

Sometimes, data is represented graphically to give a clear idea of what it represents.

Do you remember the different types of graphs which we have learnt in earlier classes?

1. A Pictograph: Pictorial representation of data using symbols.

Data Handling

CHAPTER

= 100 cars ← One symbol stands for 100 cars

July = 250 denotes

of 100

August = 300

September = ?

(i) How many cars were produced in the month of July?

(ii) In which month were maximum number of cars produced?

2. A bar graph: A display of information using bars of uniform width, their heights being proportional to

the respective values.

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38 MATHEMATICS

(i) What is the information given by the bar graph?

(ii) In which year is the increase in the number of students maximum?

(iii) In which year is the number of students maximum?

(iv) State whether true or false:

‘The number of students during 2005-06 is twice that of 2003-04.’

3. Double Bar Graph: A bar graph showing two sets of data simultaneously. It is

useful for the comparison of the data.

(i) What is the information given by the double bar graph?

(ii) In which subject has the performance improved the most?

(iii) In which subject has the performance deteriorated?

(iv) In which subject is the performance at par?

Bar heights give the

quantity for each

category.

Bars are of equal width

with equal gaps in

between.

THINK, DISCUSS AND WRITE

If we change the position of any of the bars of a bar graph, would it change the

information being conveyed? Why?

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DATA HANDLING 39

1. Month July August September October November December

Number of 1000

1500

2000

2500

1500

watches sold

2. Children who prefer School A School B School C

Walking 40 55 15

Cycling 45 25 35

3. Percentage wins in ODI by 8 top cricket teams.

Teams From Champions Last 10

Trophy to World Cup-06 ODI in 07

South Africa 75% 78%

Australia 61% 40%

Sri Lanka 54% 38%

New Zealand 47% 50%

England 46% 50%

Pakistan 45% 44%

West Indies 44% 30%

India 43% 56%

TRY THESE

Draw an appropriate graph to represent the given information.

4.2 Circle Graph or Pie Chart

Have you ever come across data represented in circular form as shown (Fig 4.1)?

The time spent by a child during a day Age groups of people in a town

Fig 4.1

(i) (ii)

These are called circle graphs. A circle graph shows the relationship between a

whole and its parts. Here, the whole circle is divided into sectors. The size of each sector

is proportional to the activity or information it represents.

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40 MATHEMATICS

For example, in the above graph, the proportion of the sector for hours spent in sleeping

number of sleeping hours

whole day

8 hours 1

24 hours 3

So, this sector is drawn as

part of the circle. Similarly, the proportion of the sector

for hours spent in school =

number of school hours

whole day

6 hours 1

24 hours 4

So this sector is drawn

of the circle. Similarly, the size of other sectors can be found.

Add up the fractions for all the activities. Do you get the total as one?

A circle graph is also called a pie chart.

TRY THESE

Fig 4.2

1. Each of the following pie charts (Fig 4.2) gives you a different piece of information about your class.

Find the fraction of the circle representing each of these information.

(i) (ii) (iii)

2. Answer the following questions based on the pie chart

given (Fig 4.3 ).

(i) Which type of programmes are viewed the most?

(ii) Which two types of programmes have number of

viewers equal to those watching sports channels?

Viewers watching different types

of channels on T.V.

4.2.1 Drawing pie charts

The favourite flavours of ice-creams for

students of a school is given in percentages

as follows.

Flavours Percentage of students

Preferring the flavours

Chocolate 50%

Vanilla 25%

Other flavours 25%

Fig 4.3

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DATA HANDLING 41

Let us represent this data in a pie chart.

The total angle at the centre of a circle is 360°. The central angle of the sectors will be

a fraction of 360°. We make a table to find the central angle of the sectors (Table 4.1).

Table 4.1

Flavours Students in per cent In fractions Fraction of 360°

preferring the flavours

Chocolate 50%

50 1

100 2

of 360° = 180°

Vanilla 25%

25 1

100 4

of 360° = 90°

Other flavours 25%

25 1

100 4

of 360° = 90°

Fig 4.4

1. Draw a circle with any convenient radius.

Mark its centre (O) and a radius (OA).

2. The angle of the sector for chocolate is 180°.

Use the protractor to draw ∠AOB = 180°.

3. Continue marking the remaining sectors.

Example 1: Adjoining pie chart (Fig 4.4) gives the expenditure (in percentage)

on various items and savings of a family during a month.

(i) On which item, the expenditure was maximum?

(ii) Expenditure on which item is equal to the total

savings of the family?

(iii) If the monthly savings of the family is ' 3000, what

is the monthly expenditure on clothes?

Solution:

(i) Expenditure is maximum on food.

(ii) Expenditure on Education of children is the same

(i.e., 15%) as the savings of the family.

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42 MATHEMATICS

(iii) 15% represents ' 3000

Therefore, 10% represents '

3000

= ' 2000

Example 2: On a particular day, the sales (in rupees) of different items of a baker’s

shop are given below.

ordinary bread : 320

fruit bread : 80

cakes and pastries : 160 Draw a pie chart for this data.

biscuits : 120

others : 40

Total : 720

Solution: We find the central angle of each sector. Here the total sale = ' 720. We

thus have this table.

Item Sales (in ') In Fraction Central Angle

Ordinary Bread 320

320 4

720 9

360 160

× ° = °

Biscuits 120

120 1

720 6

360 60

× ° = °

Cakes and pastries 160

160 2

720 9

360 80

× ° = °

Fruit Bread 80

80 1

720 9

360 40

× ° = °

Others 40

40 1

720 18

360 20

× ° = °

Now, we make the pie chart (Fig 4.5):

Fig 4.5

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DATA HANDLING 43

TRY THESE

Draw a pie chart of the data given below.

The time spent by a child during a day.

Sleep — 8 hours

School — 6 hours

Home work — 4 hours

Play — 4 hours

Others — 2 hours

THINK, DISCUSS AND WRITE

Which form of graph would be appropriate to display the following data.

1. Production of food grains of a state.

Year 2001 2002

2003 2004 2005 2006

Production 60

50 70 55 80 85

(in lakh tons)

2. Choice of food for a group of people.

Favourite food Number of people

North Indian 30

South Indian 40

Chinese 25

Others 25

Total 120

3. The daily income of a group of a factory workers.

Daily Income Number of workers

(in Rupees) (in a factory)

75-100 45

100-125 35

125-150 55

150-175 30

175-200 50

200-225 125

225-250 140

Total 480

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44 MATHEMATICS

EXERCISE 4.1

1. A survey was made to find the type of music

that a certain group of young people liked in

a city. Adjoining pie chart shows the findings

of this survey.

From this pie chart answer the following:

(i) If 20 people liked classical music, how

many young people were surveyed?

(ii) Which type of music is liked by the

maximum number of people?

(iii) If a cassette company were to make

1000 CD’s, how many of each type

would they make?

2. A group of 360 people were asked to vote

for their favourite season from the three

seasons rainy, winter and summer.

(i) Which season got the most votes?

(ii) Find the central angle of each sector.

(iii) Draw a pie chart to show this

information.

3. Draw a pie chart showing the following information. The table shows the colours

preferred by a group of people.

Colours Number of people

Blue 18

Green 9

Red 6

Yellow 3

Total 36

4. The adjoining pie chart gives the marks scored in an examination by a student in

Hindi, English, Mathematics, Social Science and Science. If the total marks obtained

by the students were 540, answer the following questions.

(i) In which subject did the student score 105

marks?

(Hint: for 540 marks, the central angle = 360°.

So, for 105 marks, what is the central angle?)

(ii) How many more marks were obtained by the

student in Mathematics than in Hindi?

(iii) Examine whether the sum of the marks

obtained in Social Science and Mathematics

is more than that in Science and Hindi.

(Hint: Just study the central angles).

Find the proportion of each sector. For example,

Blue is

18 1

36 2

; Green is

9 1

36 4

and so on. Use

this to find the corresponding angles.

Season No. of votes

Summer 90

Rainy 120

Winter 150

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DATA HANDLING 45

TRY THESE

5. The number of students in a hostel, speaking different languages is given below.

Display the data in a pie chart.

Language Hindi English Marathi Tamil Bengali Total

Number 40

12 9 7 4 72

of students

4.3 Chance and Probability

Sometimes it happens that during rainy season, you carry a raincoat every day

and it does not rain for many days. However, by chance, one day you forget to

take the raincoat and it rains heavily on that day.

Sometimes it so happens that a student prepares 4 chapters out of 5, very well

for a test. But a major question is asked from the chapter that she left unprepared.

Everyone knows that a particular train runs in time but the day you reach

well in time it is late!

You face a lot of situations such as these where you take a chance and it

does not go the way you want it to. Can you give some more examples? These

are examples where the chances of a certain thing happening or not happening

are not equal. The chances of the train being in time or being late are not the

same. When you buy a ticket which is wait listed, you do take a chance. You

hope that it might get confirmed by the time you travel.

We however, consider here certain experiments whose results have an equal chance

of occurring.

4.3.1 Getting a result

You might have seen that before a cricket match starts, captains of the two teams go out

to toss a coin to decide which team will bat first.

What are the possible results you get when a coin is tossed? Of course, Head or Tail.

Imagine that you are the captain of one team and your friend is the captain of the other

team. You toss a coin and ask your friend to make the call. Can you control the result of

the toss? Can you get a head if you want one? Or a tail if you want that? No, that is not

possible. Such an experiment is called a random experiment. Head or Tail are the two

outcomes

of this experiment.

1. If you try to start a scooter, what are the possible outcomes?

2. When a die is thrown, what are the six possible outcomes?

Oh!

raincoat.

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46 MATHEMATICS

THINK, DISCUSS AND WRITE

3. When you spin the wheel shown, what are the possible outcomes? (Fig 4.6)

List them.

(Outcome here means the sector at which the pointer stops).

4. You have a bag with five identical balls of different colours and you are to pull out

(draw) a ball without looking at it; list the outcomes you would

get (Fig 4.7).

In throwing a die:

• Does the first player have a greater chance of getting a six?

• Would the player who played after him have a lesser chance of getting a six?

• Suppose the second player got a six. Does it mean that the third player would not

have a chance of getting a six?

4.3.2 Equally likely outcomes:

A coin is tossed several times and the number of times we get head or tail is noted. Let us

look at the result sheet where we keep on increasing the tosses:

Fig 4.7

Fig 4.6

Number of tosses Tally marks (H) Number of heads Tally mark (T) Number of tails

50 | | | | | | | | | | | | 27 | | | | | | | | | | | | 23

| | | | | | | | | | | | | | | | |

60 | | | | | | | | | | | | 28 | | | | | | | | | | | | 32

| | | | | | | | | | | | | | | | | | | | | | | | |

70 ... 33 ... 37

80 ... 38 ... 42

... 44 ... 46

100 ... 48 ... 52

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DATA HANDLING 47

Observe that as you increase the number of tosses more and more, the number of

heads and the number of tails come closer and closer to each other.

This could also be done with a die, when tossed a large number of times. Number of

each of the six outcomes become almost equal to each other.

In such cases, we may say that the different outcomes of the experiment are equally

likely. This means that each of the outcomes has the same chance of occurring.

4.3.3 Linking chances to probability

Consider the experiment of tossing a coin once. What are the outcomes? There are only

two outcomes – Head or Tail. Both the outcomes are equally likely. Likelihood of getting

a head is one out of two outcomes, i.e.,

. In other words, we say that the probability of

getting a head =

. What is the probability of getting a tail?

Now take the example of throwing a die marked with 1, 2, 3, 4, 5, 6 on its faces (one

number on one face). If you throw it once, what are the outcomes?

The outcomes are: 1, 2, 3, 4, 5, 6. Thus, there are six equally likely outcomes.

What is the probability of getting the outcome ‘2’?

It is

What is the probability of getting the number 5? What is the probability of getting the

number 7? What is the probability of getting a number 1 through 6?

4.3.4 Outcomes as events

Each outcome of an experiment or a collection of outcomes make an event.

For example in the experiment of tossing a coin, getting a Head is an event and getting a

Tail is also an event.

In case of throwing a die, getting each of the outcomes 1, 2, 3, 4, 5 or 6 is an event.

← Number of outcomes giving 2

← Number of equally likely outcomes.

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48 MATHEMATICS

TRY THESE

Is getting an even number an event? Since an even number could be 2, 4 or 6, getting an

even number is also an event. What will be the probability of getting an even number?

It is

Example 3: A bag has 4 red balls and 2 yellow balls. (The balls are identical in all

respects other than colour). A ball is drawn from the bag without looking into the bag.

What is probability of getting a red ball? Is it more or less than getting a yellow ball?

Solution: There are in all (4 + 2 =) 6 outcomes of the event. Getting a red ball

consists of 4 outcomes. (Why?)

Therefore, the probability of getting a red ball is

. In the same way the probability

of getting a yellow ball =

2 1

6 3

(Why?). Therefore, the probability of getting a red ball is

more than that of getting a yellow ball.

Suppose you spin the wheel

1. (i) List the number of outcomes of getting a green sector

and not getting a green sector on this wheel

(Fig 4.8).

(ii) Find the probability of getting a green sector.

(iii) Find the probability of not getting a green sector.

4.3.5 Chance and probability related to real life

We talked about the chance that it rains just on the day when we do not carry a rain coat.

What could you say about the chance in terms of probability? Could it be one in 10

days during a rainy season? The probability that it rains is then

. The probability that it

does not rain =

. (Assuming raining or not raining on a day are equally likely)

The use of probability is made in various cases in real life.

1. To find characteristics of a large group by using a small

part of the group.

For example, during elections ‘an exit poll’ is taken.

This involves asking the people whom they have voted

for, when they come out after voting at the centres

which are chosen off hand and distributed over the

whole area. This gives an idea of chance of winning of

each candidate and predictions are made based on it

accordingly.

← Number of outcomes that make the event

← Total number of outcomes of the experiment.

Fig 4.8

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DATA HANDLING 49

2. Meteorological Department predicts weather by observing trends from the data

over many years in the past.

EXERCISE 4.2

1. List the outcomes you can see in these experiments.

(a) Spinning a wheel (b) Tossing two coins together

2. When a die is thrown, list the outcomes of an event of getting

(i) (a) a prime number (b) not a prime number.

(ii) (a) a number greater than 5 (b) a number not greater than 5.

3. Find the.

(a) Probability of the pointer stopping on D in (Question 1-(a))?

(b) Probability of getting an ace from a well shuffled deck of 52 playing cards?

4. Numbers 1 to 10 are written on ten separate slips (one number on one slip), kept in

a box and mixed well. One slip is chosen from the box without looking into it. What

is the probability of .

(i) getting a number 6?

(ii) getting a number less than 6?

(iii) getting a number greater than 6?

(iv) getting a 1-digit number?

5. If you have a spinning wheel with 3 green sectors, 1 blue sector and 1 red sector,

what is the probability of getting a green sector? What is the probability of getting a

non blue sector?

6. Find the probabilities of the events given in Question 2.

WHAT HAVE WE DISCUSSED?

1. In order to draw meaningful inferences from any data, we need to organise the data systematically.

2. Data can also be presented using circle graph or pie chart. A circle graph shows the relationship

between a whole and its part.

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50 MATHEMATICS

3. There are certain experiments whose outcomes have an equal chance of occurring.

4. A random experiment is one whose outcome cannot be predicted exactly in advance.

5. Outcomes of an experiment are equally likely if each has the same chance of occurring.

6. Probability of an event =

Number of outcomes that make an event

Total number of outcomes of the experime

, when the outcomes

are equally likely.

7. One or more outcomes of an experiment make an event.

. Chances and probability are related to real life.

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